Previously there has been a significant increase in the development and use of materials that exhibit nonlinear multi-photon behavior. These materials may be used for such applications as, e.g., a high precision medical diagnostics tools usage, effective treatments for various cancers, biological detectors (e.g., markers), three-dimensional (“3D”) micro- and/or nano-fabrication, fluorescent imaging systems, optical limiters, optical storage, semiconductor nano-sized probes, etc.
Conventional experiments that may be used to characterize the optical properties of nonlinear materials such as multi-photon organic/inorganic materials, semiconductors, fluids, gases or nanostructured materials include, e.g., z-scan procedures, optical transmission techniques, and pump-probe techniques. Facilities that are equipped to characterize such materials may require millions of dollars of equipment including, for example, lasers which can operate at different wavelengths in the ultraviolet, visible, near infrared (“IR”), mid IR and far IR regions. A laser beam can have an infinite duration (e.g., a continuous wave), or a finite duration which can be on the order of, e.g., nanoseconds (“ns”), picoseconds (“ps”), or femtoseconds (“fs”). Such facilities can also include various detectors, measurement electronics and data gathering computers that may be used to characterize these materials. A laser pulse duration or width can refer to, e.g., a continuous wave or a wave having a finite duration.
Pulse widths provided by the lasers which may be used to characterize and activate such optical materials can vary by about 12 orders of magnitude. This can make it difficult for a single numerical code to accurately and robustly characterize all possible interactions in order to reduce the need for costly experiments. Additionally, many experiments may need to be performed on a single material over many orders of magnitude of laser energies, where different electronic states of the nonlinear material can contribute to the total absorption behavior at different energy ranges. However, conventional codes may neglect higher energy levels. This simplification can yield reasonable results for particular energy ranges and incorrect results for other ranges.
Optical transmission measurements can be made using a particular laser such as, e.g., a Nd:YAG laser, a Ti:sapphire laser, a fiber laser, a semiconductor laser, a photonic crystal nanolaser, a quantum cascade laser, etc. The Nd:YAG laser can produce nanosecond pulses, whereas a Ti:sapphire laser can produce picosecond or femtosecond pulses. Each individual optical transmission measurement can be performed using a selected pulse width and a particular wavelength. However, a further measurement can be required for a different sample thickness. The number of experiments which may be required for characterizing these materials over a range of conditions and parameters can be large, and costs and time associated with such measurements can also be significant. For example, it may take many months to investigate a new material. Conventional simulation codes that can be used to model these measurements may be applicable only to a specific material interacting with a particular laser system at a certain intensity, and such codes may use simplifying assumptions that can further limit their applicability with respect to, e.g., wavelength, pulse widths, concentration of absorbing particles, and/or sample thickness. Such codes may not be capable of predicting the effects of variations in these parameters on the optical transmission behavior of a material based on one experimental measurement or a limited number of such measurements.
Conventional theoretical and/or numerical analyses of a laser beam transmission through nonlinear absorbing materials can utilize a number of assumptions that can limit their general applicability. Such nonlinear absorbing materials are described, e.g., in N. Allard et al., “The effect of neutral nonresonant collisions on atomic spectral lines,” Rev. Mod. Phys. 54, 1103-1182 (1982). Shorter pulsed lasers and multi-photon processes are becoming important in this field as described, e.g., in U. Siegner et al., “Nonlinear optical processes for ultrashort pulse generation,” in Handbook of Optics, M. Bass et al., eds., McGraw-Hill, New York, 2001, vol. IV, pp. 25-31. Thus, there may be a need for a more general approach which can increase the range of applicability of the equations used and the assumptions involved.
Conventional propagation and/or transmission analyses may neglect several molecular excited states as described, for example, in Y. R. Shen, The Principle of Nonlinear Optics, Wiley, New York, 1984. These excited states may be used to explain experimental data, particularly at high incident energy. Approximate theories of propagation and/or transmission through nonlinear materials have been formulated by various researchers in conjunction with their particular experimental data. These approximate theories may require numerical solutions, and approximate analytic expressions based on simplifying assumptions may often be used to reduce a required computational time. However, such approximate numerical solutions may not adequately describe the laser beam propagation through the material.
Additionally, because conventional approaches may often be used in conjugation with specific laser systems (e.g., with a specific wavelength and pulse duration), the resulting theoretical or numerical analysis may have a limited applicability. This approach can thus limit predictive capabilities of the analysis. For example, a theoretical description for a ns pulsed laser may not be capable of describing the effects of a ps or fs duration laser pulse interacting with the same material. Conventional theoretical or numerical analyses may provide agreement with specific experiments for specific materials and yield some insight, particularly in absorbers which may be described using single-photon processes. However, such conventional analyses may need to be modified and/or expanded to provide accurate descriptions and predictions of phenomena involving, e.g., a laser transmission through absorbers.
Changing the material or the laser beam characteristics associated with an absorption interaction may require a different numerical method and/or computer code to analyze the optical response. For example, new energy levels in the absorbing material may become accessible with an increase in laser intensity, and a new set of coupled equations may be required to describe the laser-absorber interaction. Because analytical solutions may not be possible, except in very simple cases, new computer codes may need to be written. An algorithm and/or code describing two energy levels of an absorber may not provide accurate results when three or more energy levels may contribute to a particular laser-absorber interaction. Defining new algorithms and writing new numerical codes to describe such absorption interactions can involve, e.g., months or years of effort.
Multi-photon-absorbing materials may also be used as nonlinear absorbers, including those described in, e.g., L. W. Tutt et al., “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum. Elect. 17, 299-305 (1993); J. E. Rogers et al., “Understanding the one-photon photophysical properties of a two-photon absorbing chromophore,” J. Phys. Chem. A 108, 5514-5520 (2004); J. W. Perry, “Organic and metal-containing reverse saturable absorbers for optical limiters,” in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. (Boca Raton, Fla.: CRC 1997), pp. 813-839; M. J. Potasek et al., “All optical power limiting,” J. Nonlinear Optical Physics and Materials 9, 343-365 (2000); M. J. Potasek, “High-Bandwidth Optical Networks and Communication, Photodetectors and Fiber Optics ed. H. S. Nalwa (Academic Press, 2001) pp. 459-543; D. I. Kovsh et al., “Nonlinear Optical Beam Propagation for Optical Limiting,” Appl. Opt. 38, 5168-5180 (1999); and W. Jia et al., “Optical limiting of semiconductor nanoparticles for nanosecond laser pulses,” Appl. Phys. Lett. 85, 6326-6328 (2004).
Photon absorbing materials may also be used in applications such as biological detectors as described in, e.g., S. M. Kirkpatrick et al., “Nonlinear saturation and determination of the two-photon absorption cross section of green fluorescent protein,” J. Phys. Chem. B 105, 2867-2873 (2001), and three-dimensional microfabrication procedures such as those described in, for example, S. Maruo et al., “Two-photon-absorbed near-infrared photopolymerization for three-dimensional microfabrication,” J. Microelectromechanical Systems 7, 411-415 (1998); B. H. Cumpston et al., “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature 398, 51-54 (1999); and G. Witzgall et al., “Single-shot two-photon exposure of commercial photoresist for the production of three-dimensional structures,” Opt. Let. 23, 1745-1748 (1998).
Further applications of photon absorbing materials may include fluorescent imaging systems such as those described in W. Denk et al., “Two-photon laser scanning fluorescence microscopy,” Science 248, 73-76 (1990), and optical storage systems as described, for example, in H. E. Pudavar et al., “High-density three-dimensional optical data storage in a stacked compact disk format with two-photon writing and single photon readout,” Appl. Phys. Lett. 74, 1338-1340 (1999); and in P. N. Prasad, “Emerging opportunities at the interface of photonics, nanotechnology and biotechnology,” Mol. Cryst. Liq. Cryst. 415, 1-10 (2004).
A nonlinear absorbing material in which an excited state absorption is large, as compared to a ground state absorption, can be referred to as a reversible saturable absorber (“RSA”). Such materials can exhibit a large absorption at high input laser energies, but their performance may be limited by an accompanying linear absorption at low input energy. A transparency (e.g., low absorption) at low input energy, combined with high absorption at high input energy, can be achieved with multi-photon absorber (“MPA”) materials in which two or more photons may be absorbed simultaneously. For examples, the materials that exhibit a large two-photon absorption (“TPA”) behavior may be important for a wide range of applications. Examples of TPA materials are described, for example, in M. Albota et al., “Design of organic molecules with large two-photon absorption cross sections,” Science 281, 1653-1656 (1998); and B. A. Reinhardt et al., “Highly active two-photon dyes: Design, synthesis, and characterization toward application,” Chem. Mater. 10, 1863-1874 (1998).
MPA materials can exhibit complex absorption mechanisms involving higher level electronic states. For example, TPA may be followed by excited state absorption (“ESA”) which is described, e.g., in J. Kleinschmidt et al., “Measurement of strong nonlinear absorption in stilbene-chloroform solution, explained by the superposition of two-photon absorption and one-photon absorption from the excited state,” Chem. Phys. Lett. 24, 133-135 (1974). Nonlinear transmission measurements and Z-scan measurements of organic materials can indicate the presence of ESA. These measurements are described, e.g., in D. A. Oulianov et al., “Observations on the measurements of two-photon absorption cross-section,” Opt. Comm. 191, 235-243 (2001); and S. Guha et al., “Third-order optical nonlinearities of metallotetrabenzoporphyrins and a platinum poly-yne,” Opt. Lett. 17, 264-266 (1992).
ESA can be the primary absorption mechanism in a nanosecond (ns) regime in a TPA material such as, e.g., D-π-A chromophore from the AFX group. TPA can be a primary mechanism for populating the excited states in such materials. However, TPA may dominate the total absorption behavior in the femtosecond regime. To analyze and predict the experimentally observable behavior of such materials under laser irradiation may require a solution to a nonlinear system of differential equations. Although some material systems can be described accurately by equations having a simple form which can be solved analytically, it may be important to have effective and robust numerical simulation tools to provide useful information for a wide variety of materials under a broad range of conditions.
For many RSA and TPA materials such as those described, e.g., in G. S. He et al., “Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,” J. Chem. Phys. 120, 5275-5284 (2004); and R. Kannan et al., “Toward highly active two-photon absorbing liquids. Synthesis and characterization of 1,3,5-triazine-based octupolar molecules,” Chem. Mater. 16, 185-194 (2004), simulation calculations can be based on a solution of a coupled system of propagation and rate equations. The rate equations may be formulated using a phenomenological five-level absorption model which is described, for example, in R. L. Sutherland et al., “Excited state characterization and effective three-photon absorption model of two-photon-induced excited state absorption in organic push-pull charge-transfer chromophores,” J. Opt. Soc. Am. B 22, 1939-1948 (2005).
The propagated light in the RSA materials may attenuate as a result of electron excitations from the ground state and from singlet and/or triplet excited states. The absorption mechanism in the TPA materials can be similar to that in RSA materials, except that two photons can be absorbed during a transition from the ground state to the first singlet excited state. Depending on the pulse width and intensity of the incident light, the electron population densities may change which can alter the transmittance characteristics of the material. Solving equations describing light propagation in three-photon absorption (“3PA”) materials such as, e.g., PPAI, which is described, e.g., in D.-Y. Wang et al., “Large optical power limiting induced by three-photon absorption of two stibazolium-like dyes,” Chem. Phys. Lett. 369, 621-626 (2002), may be less problematic because the absorption model may include just two levels. In such materials, an incident pulse intensity may decrease due to simultaneous absorption of three photons from the ground level to the lowest singlet excited state. However, experimental investigations of 3PA materials are in an early stage and more complex nonlinear absorption models should be used for these materials.
Numerical methods may often be used to solve coupled equations describing laser-matter interactions, because there are few analytic solutions for such equations. New numerical code may be written to describe each energy level diagram representing a particular material of interest and an associated laser interaction. Such codes can vary in their degree of sophistication and in any approximations used, which may limit their applicability to certain lasers, as well as to particular temporal and/or radial domains. New numerical codes may be required to describe an increasing variety of possible interactions between lasers and materials. For example, a large number of individual computer codes have been written to solve various approximate sub-sets of laser-material interactions. As more lasers are developed having new wavelengths and/or pulse widths, many additional codes or modifications of existing codes may need to be written to describe them quantitatively.
Thus, there may be a need overcome the above-described deficiencies and issues to facilitate the effective and robust numerical simulation tools to measure, analyze, and predict the behavior of photon absorbing materials that are exposed to a laser irradiation. Further, there may be a need for a uniform solver which is capable of modeling a variety of nonlinear materials having different absorption configurations under a range of the irradiation conditions such as, e.g., different wavelengths, pulse widths, sample thicknesses, etc. Such exemplary simulation tools may provide guidelines for developing new functional materials, e.g., for designing molecular or semiconductor quantum dots or wires that may reduce development costs. The numerical method or computer program for such a simulation tool may not need to be rewritten when the material or laser conditions are changed.